Kähler Metrics of Constant Scalar Curvature on Hirzebruch Surfaces

Transcription

1 ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for athematical Physics A-1090 Wien, Austria Kähler etrics of Constant Scalar Curvature on Hirzebruch Surfaces Hiroyuki Kamada Vienna, Preprint ESI 1766 (2006) January 23, 2006 Supported by the Austrian Federal inistry of Education, Science and Culture Available via

2 Kähler metrics of constant scalar curvature on Hirzebruch surfaces Hiroyuki Kamada Abstract It is shown that a Hirzebruch surface admits a Kähler metric (possibly indefinite) of constant scalar curvature if and only if its degree equals zero. There have been many extensive studies for positive-definite Kähler metrics of constant scalar curvature, especially, Kähler Einstein metrics and scalar-flat Kähler metrics, on existence, uniqueness, obstructions, and relationships with other notions, for example, certain stabilities of polarized Kähler manifolds (e.g., Donaldson [7] and references therein). An indefinite counterpart of the notion of Kähler metrics of constant scalar curvature is defined in a natural way similar to that in the positive-definite case, and includes the notions of indefinite Kähler-Einstein metrics and scalarflat indefinite Kähler metrics. The existence problem for such metrics has its own task in the geometry of pseudo-riemannian manifolds. The lowest real dimension of an indefinite Kähler manifold (, g) must be four and the signature of the metric g must be (2, 2). atsushita [20] studied the existence problem for metrics of signature (2, 2), from a point of view of that for fields of two-planes. Indefinite Kähler metrics of signature (2, 2) also appear in mathematical physics. otivated by the work of Ooguri-Vafa [23] on string theory, Petean [24] studied the existence problem for indefinite Kähler Einstein metrics on compact complex surfaces and obtained a complete classification of compact complex surfaces that admit Ricci-flat indefinite Kähler metrics. On the other hand, concerning the existence of scalar-flat indefinite Kähler metrics on compact complex surfaces (or equivalently, self-dual Kähler metrics of neutral signature), several results have been known. For example, many scalar-flat (non- Ricci-flat) indefinite Kähler metrics have been constructed explicitly on the product S 2 S 2 of two two-dimensional spheres, by an indefinite analogue of LeBrun s ansatz (Tod [26], Kamada [14]). By construction, each of these metrics has an obvious (Hamiltonian) S 1 -symmetry. Conversely, it has been shown in [14] that a compact scalar-flat indefinite Kähler surface (more generally, a 2000 athematics Subject Classification: 14J26, 53C55, 53C50. Key words and phrases: Hirzebruch surfaces, Kähler metrics, constant scalar curvature, Bando-Calabi-Futaki obstruction. Supported by EXT. Grant-in-Aid for Young Scientists

3 compact indefinite Kähler surface with vanishing total scalar curvature) that admits a Hamiltonian S 1 -symmetry must be biholomorphic to a Hirzebruch surface F d (d = 0, 1, 2,...). In fact, F d admits a scalar-flat indefinite Kähler metric if and only if the degree d equals zero, without any assumption on symmetry. It is then natural to ask whether a similar result holds for (indefinite) Kähler metrics of constant scalar curvature on F d. In this article, we will discuss the existence problem for Kähler metrics of constant scalar curvature on a Hirzebruch surface F d, from a unified aspect in both definite (i.e., positive- or negative-definite) case and indefinite case. It is clear that F 0 is biholomorphic to the product P 1 P 1 of two complex projective lines and admits not only definite but also indefinite Kähler metrics of constant scalar curvature, given by a product metric on F 0 = P 1 P 1. So the problem is reduced to the case d 1. In the definite case, there are two famous obstructions to the existence of Kähler metrics of constant scalar curvature: the atsushima-lichnerowicz obstruction (atsushima [19], Lichnerowicz [16]) and the Bando-Calabi-Futaki obstruction (Bando [1], Calabi [5], Futaki [9, 10]). The author does not know whether the former obstruction can be generalized to indefinite cases; however, the latter can fortunately be done in a certain situation (Futaki-abuchi [11], cf. [14]). As seen in [14], this generalization of the Bando-Calabi-Futaki obstruction is defined on certain compact complex manifolds, when the Kähler classes (possibly indefinite) are integral. In this paper, we give a definition of the Bando-Calabi-Futaki obstruction for any indefinite Kähler class, under additional assumptions on the underlying complex manifold. As an application of this obstruction, we prove the following result: Theorem 1 A Hirzebruch surface F d admits a Kähler metric (possibly indefinite) of constant scalar curvature if and only if d = 0. This paper is organized as follows: In Section 1, we will explain a generalization of the Bando-Calabi-Futaki obstruction for a compact possibly indefinite Kähler manifold. Section 2 is devoted to introducing Nakagawa s formula (Nakagawa [21]) of the Bando-Calabi-Futaki obstruction for compact toric surfaces. In Section 3, we will recall basic properties of a Hirzebruch surface F d and its toric realization, and will prove Theorem 1, by computing the Bando-Calabi- Futaki obstruction for possible Kähler classes on F d. 1 Bando-Calabi-Futaki obstruction Let be a compact complex manifold of complex dimension n, and g a (possibly indefinite) Kähler metric on with Kähler form ω. In this article, we will also use the words Kähler surfaces, Kähler metrics and Kähler forms, etc., even in indefinite cases. When n = 2, the metric g is definite or indefinite according as ω 2 := ω ω > 0 or ω 2 < 0 with respect to the complex orientation of. 2

4 In any dimension, the Ricci form γ of a Kähler manifold (, g) is a d- closed real (1, 1)-form on given by γ = 1 log det(g α β), and the scalar curvature s of (, g) is characterized by the equation γ ω n 1 = (s/2n)ω n. A Kähler manifold (, g) is said to be of constant scalar curvature, if s is constant. Let (, g) be a compact Kähler manifold (possibly indefinite) with Kähler form ω and 2πη := [ω] bc denote its Bott-Chern class, which is referred to as the Kähler class of ω in this paper. Here, the Bott-Chern class [a] bc of a d-closed real (1, 1)-form a means an element in the Bott-Chern cohomology group H 1,1 bc(; R) = Ker(d : Ω1,1 (; R) Ω 2,1 () + Ω 1,2 ()) Im( 1 : Ω 0,0 (; R) Ω 1,1 (; R)) determined by a (Bott-Chern [4], Barth et al.[2]), where Ω p,q () and Ω p,p (; R) denote the space of (p, q)-forms and that of real (p, p)-forms on, respectively. If admits a positive-definite Kähler metric, then the Bott-Chern class [a] bc of a real (1, 1)-form a is determined by its de Rham cohomology class [a] dr, via the global -lemma. In general, for a compact complex manifold, the following short exact sequence exists: 0 H 1 (; O)/H 1 (; R) H 1,1 bc(; R) H 1,1 dr(; R) 0, (1) where H 1 (; O) denotes the first cohomology group of with coefficients in the structure sheaf O = O of. Note that H 1 (; O) is isomorphic to the Dolbeault cohomology group H 0,1 (). It then follows that the (global) lemma holds for real (1, 1)-forms on if and only if the first Betti number b 1 () of is the twice of its irregularity q() = dim C H 1 (; O), that is, b 1 () = 2q() (Gauduchon [12]). In this case, dropping the indications BC and DR, we may write [ ] bc and [ ] dr simply as [ ]. On a compact complex surface, the -lemma holds for real (1, 1)-forms if and only if admits a definite Kähler metric (Barth et al.[2]). Usually, the (real) first Chern class c 1 (L) of a complex line bundle L over a complex manifold is defined as an element of the second cohomology group H 2 dr(; Z) H 2 dr(; R). Suppose also that L is a holomorphic line bundle over. Taking a Hermitian fiber-metric h L on L, we can express the first Chern class c 1 (L) of L as c 1 (L) = ( 1/2π)[ log h L ] dr. Let H 1,1 dr(; R) be the set of de Rham cohomology classes each of which has a representative by a real closed (1, 1)-form on, and set H 1,1 dr(; Z) := H 1,1 dr(; R) H 2 dr(; Z). Then it is well-known that each element in H 1,1 dr(; Z) can conversely be represented by the first Chern class c 1 (L) of a holomorphic line bundle L over. oreover, for a holomorphic line bundle L over, we can define an element ĉ 1 (L) in H 1,1 bc(; R) by ĉ 1 (L) := ( 1/2π)[ logh L ] bc. It is known that ĉ 1 (L) is independent of the choice of h L and is also called the first Chern class of L. In particular, letting K 1 be the anti-canonical bundle of, we call ĉ 1(K 1 ) and c 1 (K 1 ) the first Chern class of, and denote them by ĉ 1() and c 1 (), respectively. Let be a compact complex manifold and h() the set of all holomorphic vector fields on. For α H 1,1 bc(; R) and V h(), we define a pairing 3

5 α, V by H 1,1 bc(; R) h() H 0,1 () = H 1 (; O) (α, V ) α, V := [V a], where means the interior derivative, [ ] denotes the Dolbeault cohomology class and a is any representative of α with [a] bc = α. Then the value of a pairing α, V in H 1 (; O) is independent of the choice of a in α. Now we set C := {(α, V ) H 1,1 bc(; R) h() α, V = 0 in H 1 (; O)}. Note that, if q() = 0, then C = H 1,1 bc(; R) h(). We denote the projections from H 1,1 bc(; R) h() to the first factor H 1,1 bc(; R) and to the second factor h() and their restrictions to C, by p 1 and p 2, respectively. For each α = [a] bc H 1,1 bc(; R), an element V in p 2 (p 1 1 (α)) satisfies V a = v for some smooth function v, which is called a holomorphy potential of V with respect to a. Note that a holomorphy potential v of V with respect to a in α is determined uniquely up to an additive constant. Furthermore, h α () := p 2 (p1 1 (α)) is not only a linear subspace but also a Lie subalgebra of h(). Indeed, let v and w be holomorphy potentials for arbitrary V and W in h α () with respect to a representative a of α (i.e., V a = v and W a = w), respectively. Then we have [V, W] a = (Wv V w). This shows that h α () is a Lie subalgebra of h(). Suppose that admits a Kähler metric g (possibly indefinite) and let K be the set of all Kähler classes on. For 2πη K, an element V in h η () = p 2 (p1 1 (η)) is called a Hamiltonian holomorphic vector field on (, 2πη). If q() = 0, then every holomorphic vector field is Hamiltonian holomorphic with respect to an arbitrary Kähler class on. Let ω be a Kähler form (possibly indefinite) on and V a Hamiltonian holomorphic vector field on (, [ω] bc ) satisfying V ω = v. Set / ˆF (ω, v) := n v (γ ω n 1 µ ω n ) ω n, (2) where µ := ĉ 1 () η n 1 /η n is called the averaged total scalar curvature of (, g). It is immediate that ˆF (ω, v) is independent of the choice of v. Hence we may write it as ˆF (ω, V ). Let K be the set of Kähler forms on of all possible signature. Set D := (K h()) C and D := {(ω, V ) K h() ([ω] bc, V ) D }. Then (2) defines a map ˆF : D C. In what follows, we explain that, under certain assumptions, ˆF : D C descends to a map ˆF : D C, that is, ˆF (ω, V ) depends only on [ω] bc rather than ω. To do this, we set F (ω, v) := n (2π) n v(γ ω n 1 µ ω n ). 4

6 Since F (ω, v) is independent of the choice of a holomorphy potential v of V, we may also denote it by F (ω, V ). If g is positive-definite, then F (ω, ) is independent of the choice of ω in 2πη and coincides with the restriction of the usual Bando-Calabi-Futaki character, which is written as F η, to hη (). If η = c 1 (L) for an ample line bundle L over and if a holomorphic vector field V on has a holomorphic lifting Ṽ onto L exists, then there is a famous formula for F c1(l) (V ), known as Tian s formula ([25]), in terms of the data of L, K 1 and Ṽ. Indeed, F (ω, V ) is expressed as F (ω, V ) = 2πnµ n + 1 F c L n+1 1 (Ṽ ) where F cn+1 1 (Ṽ ) is defined by E F cn+1 1 E (Ṽ ) := 2π 2 n (n + 1)! j=0 n ( ) ( 1) j n j F cn+1 1 K 1 ( ) n+1 1 ( θ(h)(ṽ ) + Θ(h)) n+1 2π Ln 2j(Ṽ ), (3) for E = L or E = K 1 Ln 2j (j = 0,..., n). Here, h is a Hermitian fibermetric on E, and θ(h) and Θ(h) are the connection form and the curvature form of (E, h), respectively. Once a lifting Ṽ of V onto L is chosen, the lifting Ṽ of V onto each K 1 Ln 2j is naturally determined, and moreover F cn+1 1 E (Ṽ ) is independent of the choice of a Hermitian fiber-metric h on E, since arbitrary two Hermitian fiber-metrics on E can be joined by a smooth family of Hermitian fiber-metrics. Even in the case where g is indefinite, Tian s formula is also available, by taking detg as a Hermitian fiber-metric on K 1. Since two Kähler forms ω and ω with the same Kähler class 2πĉ 1 (L) determine Hermitian fiber-metrics detg and detg on K 1 and each term in the right hand side of (3) is independent of the choice of a Hermitian fiber-metric, F (ω, V ) and hence ˆF (ω, V ) are independent of the choice of ω in 2πĉ 1 (L), under the assumption of the existence of a holomorphic lifting Ṽ of V onto L (cf. [14]). As shown in Kobayashi [15], any Hamiltonian holomorphic vector field V on a Hodge manifold can lift holomorphically onto any holomorphic line bundle E over. By observing its proof, one can obtain the following result, which is also pointed out essentially in Donaldson [7, 8]: Proposition 1 Let be a complex manifold and (2πα, V ) an element in C, that is, α has a representative a with V a = v for some smooth function v on. Suppose also that α is integral, that is, α = ĉ 1 (E) for some holomorphic line bundle E over. Then V has a holomorphic lifting Ṽ onto E. Let us define D int and D int respectively by D int := {(2πĉ 1 (E), V ) D E is a holomorphic line bundle over }, D int := {(ω, V ) D ([ω] bc, V ) D int }. 5

7 As explained above, F (ω, V ) and ˆF (ω, V ) are independent of the choice of ω in 2πη for any element (2πη, V ) in D int. Namely, F, ˆF : D int C descend to functions on D int, which we denote respectively by F, ˆF : D int C. As in the positive-definite case, we also use the notations F η (V ) := F (2πη, V ) and ˆF η (V ) := ˆF (2πη, V ) for (2πη, V ) D int. Recall that R := R \ {0} acts naturally on D by κ (2πη, V ) := (2πκη, V ) for κ R. Let v(ω, V ) denote a holomorphy potential of V with respect to ω K. For any κ R, a holomorphy potential v(κω, V ) of V with respect to κω is given by κv(ω, V ). Then we have F (κω, κv) = κ n F (ω, v), ˆF (κω, κv) = ˆF (ω, v). Thus, for any (2πη, V ) R D int := {κ (2πηint, V ) D (2πη int, V ) D int }, one can verify that ˆF (ω, V ) and F (ω, V ) are independent of the choice of ω in 2πη. Hence we denote them by F (2πη, V ) and ˆF (2πη, V ), respectively. If 2πη also includes a Kähler form (possibly indefinite) of constant scalar curvature, then F (2πη, V ) and ˆF (2πη, V ) must vanish. Definition 1 F : R D int C and ˆF : R D int /R C are called the Bando-Calabi-Futaki obstruction and the normalized Bando-Calabi-Futaki obstruction for the existence of Kähler metrics of constant scalar curvature, respectively. In the remainder of this section, we try to generalize the Bando-Calabi- Futaki obstruction for non-integral Kähler classes. Lemma 1 Let be a compact complex manifold endowed with (2πη, V ) in D and (ω, v) a pair in (2πη K ) C (; C) with V ω = v. Suppose that there exists a sequence {(ω j, v j, m j )} in K C (; C) (Z \ {0}) such that (i) (m j [ω j ] bc, V ) D int with V ω j = v j (j = 1, 2,...) (ii) {(ω j, v j )} converges to (ω, v) as j in C 2 -topology. Then F (ω, V ) and ˆF (ω, V ) are independent of the choice of ω in 2πη. Thus, in the situation in Lemma 1, we may write F (ω, V ) and ˆF (ω, V ) as F (2πη, V ) and ˆF (2πη, V ), and also call F : D C and ˆF : D /R C the Bando-Calabi-Futaki obstruction and the normalized Bando- Calabi-Futaki obstruction, respectively. Proof of Lemma 1. Let γ j and γ denote the Ricci forms of ω j and ω, respectively. Then γ j γ as j, since γ j = 1 log det(g j ) = γ + 1 log ω n j /ω n. Then it follows that lim F (ω j, v j ) = lim v j (γ j ω n 1 j µ j ωj n ) = v(γ ω n 1 µ ω n ) = F (ω, v). 6

8 Thus we may express this fact as F (ω, V ) = lim F (ω j, V ). Let ω and ω be two Kähler forms (possibly indefinite) on with the same Kähler class [ω] bc = [ω ] bc = 2πη, Then there exists a smooth real function ϕ on such that ω ω = 1 ϕ. Define ω j := ω j + 1 ϕ and v j := v j + 1V ϕ. Clearly, [ω j ] bc = [ω j ] bc = (2π/m j )ĉ 1 (L j ) for a holomorphic line bundle L j over, V ω j = v j (j = 1, 2,...) and (ω j, V ) D int for any sufficiently large j. Since (ω j, v j) (ω, v := v + 1V ϕ) as j holds, {(ω j, v j )} satisfies properties similar to those for {(ω j, v j )}. Therefore we have ˆF (ω, V ) = lim ˆF (ω j, V ) = lim ˆF (m j ω j, V ) = lim ˆF (ĉ 1 (L j ), V ) = lim ˆF (ω j, V ) = ˆF (ω, V ), so ˆF (ω, V ) and hence F (ω, V ) are independent of the choice of ω in 2πη. Remark 1 Let ω and ω be arbitrary two Kähler forms in the same Kähler class 2πη on. If these forms are positive-definite, we can join ω and ω by a family {ω t } of positive-definite Kähler forms on with the same Kähler class 2πη, for example, by the line segment ω t = (1 t)ω + tω. Then we can easily prove that d F (ω t, V )/dt 0, which implies F (ω, V ) = F (ω, V ). However, when ω and ω are indefinite, it is not clear at all whether we can join these forms by such a family {ω t }. For example, even though the line segment ω t = (1 t)ω + tω is nondegenerate for t sufficiently near t = 0 or 1, it may be degenerate for some t (0 < t < 1). If we can fortunately join ω and ω by a smooth family of Kähler forms {ω t } in 2πη, then we can prove F (ω, V ) = F (ω, V ) for any V h η () in the same way as that in the positive-definite case. Hence, F and ˆF descend to maps on D and D /R, respectively. In this situation, ω and ω clearly have the same signature. Concerning the signature of Kähler forms in general, we can show the following result (cf. aschler [17, 18]), by using the asymptotic versions of the (weak) orse inequality and Riemann-Roch formula in D ly [6]. Proposition 2 Let be a compact Kähler manifold (possibly indefinite) with Kähler class 2πη. Suppose that Kähler forms ω and ω in 2πη are approximated by sequences {ω j } and {ω j } consisting of rational Kähler forms on satisfying [ω j ] bc = [ω j ] bc (j = 1, 2,...). Then ω and ω have the same signature. If we can find an approximation {ω j } consisting of rational Kähler forms ω j on for each element ω in K, then the signature of a Kähler class does make sense, and K can be decomposed into the following fashion: K = p+q=n K p,q (disjoint union), where K p,q denotes the set of all Kähler classes on of signature (2p, 2q). Taking account of the proof of Proposition 3, we can verify that such an approximation {ω j } above exists for each Kähler form ω on if H 2 (; O) = {0}. 7

9 Once we can find an approximation as in Lemma 1 for each Kähler form in a given Kähler class 2πη, the obstructions F (2πη, V ) and ˆF (2πη, V ) are well-defined. In the following situation, the existence of such approximations is insured: Proposition 3 Let be a compact Kähler manifold (possibly indefinite) with Kähler form ω. Suppose that H q (; O) = {0} (q = 1, 2). Let V be any (Hamiltonian) holomorphic vector field on with a holomorphy potential v with respect to ω. Then there exists a sequence {(ω j, v j, m j )} K C (; C) (Z \ {0}) satisfying the conditions (i) and (ii) in Lemma 1. Proof. Putting H 1 (; O) = {0} in (1), we see that the -lemma holds for real (1, 1)-forms on, so we may abbreviate [ ] bc and ĉ 1 to [ ] and c 1, respectively. Since is compact, the image of c 1 : H 1 (; O ) H 2 (; Z) coincides with H 1,1 (; Z), where O denotes the sheaf of germs of nonvanishing holomorphic functions on, and the following exact sequence exists: 0 H 1 (; Z) H 1 (; O) H 1 (; O ) H 2 (; Z) H 2 (; O). Plugging H 1 (; O) = H 2 (; O) = {0} into the sequence above, we have b 1 () = 0 and H 1 (; O ) = H 2 (; Z). Then H 1,1 (; Z) = H 2 (; Z) holds. Let ψ 1,..., ψ l be d-closed real (1, 1)-forms on whose de Rham classes [ψ 1 ],..., [ψ l ] form a basis for Hdr 1,1 (; Z) = Hdr(; 2 Z). Then [ψ 1 ],..., [ψ l ] also form a basis for Hdr(; 2 R) (thus l = b 2 ()), so the Kähler class [ω] is expressed as [ω] = 2π l ν=1 aν [ψ ν ] for real numbers a ν (ν = 1,..., l). Thus ( l ω = 2π a ν ψ ν + ) 1 f ν=1 holds for some real smooth function f on. Since H 1 (; O) = {0}, it follows that V ψ ν = u ν for some smooth function u ν (ν = 1,..., l). Furthermore, ( l v 0 = 2π a ν u ν + ) 1V f ν=1 is a holomorphy potential of V with respect to ω. Then v v 0 + c for some constant c. For each ν, take a sequence {a ν j} j=1 consisting of rational numbers such that a ν j a ν as j. Set ( l ω j := 2π a ν jψ ν + ) ( l 1 f and v j := 2π a ν ju ν + ) 1V f + c. ν=1 Let {m j } be a sequence consisting of integers m j such that m j a 1 j,..., m j a l j are also integers (j = 1, 2,...). Then there exists a holomorphic line bundle L j such that 2πc 1 (L j ) = m j [ω j ] (j = 1, 2,...). For sufficiently large j, this sequence {(ω j, v j, m j )} satisfies the required conditions. ν=1 8

10 2 Toric surfaces and Nakagawa s formula For compact toric manifolds, Nakagawa [21] obtained a combinatorial formula of the Bando-Calabi-Futaki character, by making use of Tian s formula [25] and Bott s residue formula [3]. It is well-known that all compact toric surfaces are projective and hence a Hodge manifold. Then the Bando-Calabi-Futaki character for any integral (possibly indefinite) Kähler class is well-defined on a compact toric surface. Then Nakagawa s formula is also available, even in the indefinite case. In this section, we recall Nakagawa s combinatorial formula for compact toric surfaces. Let T 2 := (C ) 2 be a two-dimensional algebraic torus, (t 1, t 2 ) the standard holomorphic coordinates for T 2, and {τ 1, τ 2 } a basis for the Lie algebra t 2 of T 2 given by τ i := t i / t i (i = 1, 2). Let X Σ be a compact toric surface associated with a complete nonsingular fan Σ in N := Z 2 and Σ(i) denote the set of i-dimensional cones in Σ (i = 0, 1, 2). Then T 2 acts biholomorphically on X Σ, which has an open dense T 2 -orbit isomorphic to T 2. Note that t 2 is regarded as a complex Lie subalgebra of h(x Σ ). Then, for any S GL 2 (C), we define V 1 (S), V 2 (S), which we also regard as holomorphic vector fields on X Σ, by (V 1 (S), V 2 (S)) = (τ 1, τ 2 )S. For each σ Σ(2) and S GL 2 (C), we set A(σ) := ( a 1 (σ), a 2 (σ)) GL 2 (Z) and Q(S; σ) := A(σ) 1 S GL 2 (C), where a 1 (σ), a 2 (σ) N form a generator of σ. Let a i (σ) Σ(1) be the one-dimensional cone R 0 a i (σ) generated by a 1 (σ), a 2 (σ) N. For a map α : Σ(1) Z, we set β(s; σ, α) := (β 1 (S; σ, α), β 2 (S; σ, α)) = α(σ)q(s; σ), where α(σ) := (α( a 1 (σ) ), α( a 2 (σ) )). Let D ν denote the T 2 -invariant divisor corresponding to ν Σ(1). Then a map α : Σ(1) Z defines a T 2 -invariant divisor D(α) := ν Σ(1) α(ν)d ν on X Σ. By L α = O(D(α)) we denote the holomorphic line bundle over X Σ corresponding to D(α). For example, the canonical bundle K XΣ of X Σ corresponds to the map α : Σ(1) Z defined by α(ν) 1 for all ν Σ(1). Then we can state the following formula for toric surfaces (see [21]). Proposition 4 Let X Σ be a compact toric surface associated with a complete nonsingular fan Σ, and S GL 2 (C) a nondegenerate matrix, that is, all the components q j i(s; σ) of Q(S; σ) never vanish. For a map α : Σ(1) Z, we have c 1 F 1(L α) X Σ (V i (S)) = β i (S; σ, α) ( 2 q 1 i(s; σ) + q 2 i(s; σ) ) q 1 σ Σ(2) i(s; σ)q 2 (4) i(s; σ) 2 3 µ α (i = 1, 2), where µ α = c 1 (X Σ ) c 1 (L α )/c 2 1 (L α). σ Σ(2) β i (S; σ, α) 3 q 1 i(s; σ)q 2 i(s; σ) We can also express µ α in terms of β i (S; σ, α) and q j i(s; σ) explicitly, but omit it here (see [21]). Since F c1(lα) X Σ is C-linear, we have (F c1(lα) X Σ (τ 1 ), F c1(lα) (τ 2 )) = (F c1(lα) (V 1 (S)), F c1(lα) X Σ (V 2 (S)))S 1. X Σ X Σ 9

11 3 Hirzebruch surfaces A Hirzebruch surface F d is realized as a complex submanifold in the product P 2 P 1 of the complex projective plane P 2 and the complex projective line P 1 : F d = {(w 0 : w 1 : w 2, z 0 : z 1 ) P 2 P 1 z d 0w 2 = z d 1w 1 } (d = 0, 1, 2,...) (Hirzebruch [13], Barth et al.[2]). Then F d admits a positive-definite Kähler metric induced from a product metric on P 2 P 1. Via the projection to the second factor P 1, we can also regard F d as a P 1 -bundle P(O(d) O) P 1. Here, O(d) and O denote the holomorphic line bundle of degree d and the trivial line bundle over P 1, respectively. Note that q(f d ) = dim C H 1 (F d ; O) = 0 and p g (F d ) = dim C H 2 (F d ; O) = 0. oreover, F d is a typical example of a compact toric surface. Indeed, (λ, µ) (w 0 : w 1 : w 2, z 0 : z 1 ) := (w 0 : µw 1 : λ d µw 2, z 0 : λz 1 ). gives a holomorphic T 2 = (C ) 2 -action on F d. To apply Proposition 4 to our problem, we next realize each Hirzebruch surface F d as a compact toric surface associated with a complete nonsingular fan (d = 0, 1, 2,...)(cf. Oda [22]). Let N := Z 2 and N R := N Z R, and set v 0 = v 4 = e 1 (:= t (1, 0)), v 1 = e 2 (:= t (0, 1)), v 2 = e 1 + d e 2, v 3 = e 2. Define ν i and σ i by ν i := R 0 v i and σ i := R 0 v i + R 0 v i+1, and let A(σ k ) = ( a 1 (σ k ), a 2 (σ k )) = (a i j(σ k )) be a 2 2-matrix defined by A(σ i ) = ( v i, v i+1 ) (i = 0, 1, 2, 3). Then Σ := {0, ν 0, ν 1, ν 2, ν 3, σ 0, σ 1, σ 2, σ 3 } is a complete nonsingular fan, and the corresponding toric surface X Σ is known to be a toric realization of F d. Note here that the holomorphic vector fields τ 1 and τ 2 associated to the T 2 -action on F d are indeed given by τ 1 = / λ (λ,µ)=(1,1) and τ 2 = / µ (λ,µ)=(1,1). Let D νi be the T 2 -invariant divisor corresponding to ν i. Then the intersection numbers D νi D νj (0 i j 3) are given by D 2 ν 1 = D 2 ν 3 = d, D 2 ν 0 = D 2 ν 2 = D ν0 D ν2 = D ν1 D ν3 = 0, D νi D νj = 1 (otherwise). We express the linear equivalence classes (or the Poincaré duals) of the divisors D ν0,..., D ν3 as [D ν0 ],..., [D ν3 ], respectively. Then [D ν0 ] and [D ν1 ] are linearly independent, and generate H 2 (F d ; R), since b 2 (F d ) = 2. Let L be any holomorphic line bundle over F d. We may assume that L = L α corresponds to the divisor D α = 3 k=0 α kd νk for an α : Σ(1) Z, α(ν k ) = α k. Note that α is not unique for c 1 (L). Indeed, we have c 1 (L α ) = 3 α k [D νk ] = (α 0 + α 2 + α 3 d)[d ν0 ] + (α 1 + α 3 )[D ν1 ], (5) k=0 since [D ν2 ] = [D ν0 ] and [D ν3 ] = [D ν1 ] + d[d ν0 ]. Then we see that c 2 1(L α ) = (α 1 + α 3 )[2(α 0 + α 2 + α 3 d) (α 1 + α 3 )d], (6) c 1 (K 1 F d ) c 1 (L α ) = 2(α 0 + α 2 + α 3 d) (α 1 + α 3 )(d 2), (7) 10

12 and hence obtain µ α = 2(α 0 + α 2 + α 3 d) (α 1 + α 3 )(d 2) (α 1 + α 3 )[2(α 0 + α 2 + α 3 d) (α 1 + α 3 )d]. (8) If c 1 (L α ) contains an indefinite (resp. positive-definite) Kähler form ω, then (α 1 + α 3 )[2(α 0 + α 2 + α 3 d) (α 1 + α 3 )d] < 0 (resp. > 0). (9) By definition, ( we see) that α(σ i ) = (α i, α i+1 ) (i = 0, 1, 2, 3), where α 4 = α Setting S := π π 2 and computing Q(S; σ i ) = A(σ i ) 1 S, we have β(s; σ 0, α) = α(σ 0 )Q(S; σ 0 ) = (α 0 + πα 1, α 0 + π 2 α 1 ), β(s; σ 1, α) = α(σ 1 )Q(S; σ 1 ) = ((α 1 d α 2 ) + πα 1, (α 1 d α 2 ) + π 2 α 1 ), β(s; σ 2, α) = α(σ 2 )Q(S; σ 2 ) = (( α 2 α 3 d) πα 3, ( α 2 α 3 d) π 2 α 3 ), β(s; σ 3, α) = α(σ 3 )Q(S; σ 3 ) = (α 0 πα 3, α 0 π 2 α 3 ). By computing the terms in (4), we have c 1F 1(L α) F d (V i (S)) 3 β i (S; σ k, α) ( 2 q 1 i(s; σ k ) + q 2 i(s; σ k ) ) = = 2 3 µ α q 1 k=0 i (S; σ k)q 2 i (S; σ k) k=0 ( (α0 + π i α 1 ) 2 (1 + π i ) π i + [(α 1d α 2 ) + π i α 1 ] 2 ( 1 + d + π i ) (d + π i ) 3 β i (S; σ k, α) 3 q 1 i (S; σ k)q 2 i (S; σ k) + [( α 2 α 3 d) π i α 3 ] 2 ( 1 d π i ) + (α 0 π i α 3 ) 2 (1 π i ) ) d + π i π ( ) i 2(α0 + α 2 + α 3 d) (α 1 + α 3 )(d 2) 2 3 2(α 0 + α 2 + α 3 d) (α 1 + α 3 )d ( (α0 + π i α 1 ) 3 + [(α 1d α 2 ) + π i α 1 ] 3 π i (d + π i ) + [( α 2 α 3 d) π i α 3 ] 3 + (α 0 π i α 3 ) 3 d + π i π i = (α 1 + α 3 ) 2 {(α 1 + α 3 ) [2(α 0 + α 2 + α 3 d) (α 1 + α 3 )d]} (d 2 + 2π i d) 3[2(α 0 + α 2 + α 3 d) (α 1 + α 3 )d] (i = 1, 2). Thus we obtain (F c1(lα) F d ( 1τ 1 ), F c1(lα) F d ( 1τ 2 )) = (α 1 + α 3 ) 2 {(α 1 + α 3 ) [2(α 0 + α 2 + α 3 d) (α 1 + α 3 )d]} (d 2, 2d). (10) 3[2(α 0 + α 2 + α 3 d) (α 1 + α 3 )d] By using this formula (10), we can prove our main result. ) 11

13 Theorem 1 F d admits a Kähler metric (possibly indefinite) of constant scalar curvature if and only if d = 0. Proof of Theorem 1. As already mentioned, each Kähler class of a definite or indefinite Kähler metric on F 0 = P 1 P 1 contains a Kähler metric of constant scalar curvature. From now on, we suppose d 1. If c 1 (L α ) contains the Kähler form of a Kähler metric of constant scalar curvature, then F c1(lα) F d 0. By (9) and (10), we have 2(α 0 + α 2 + α 3 d) (α 1 + α 3 )d = α 1 + α 3. (11) Plugging (11) into (6), we obtain c 2 1(L α ) = (α 1 + α 3 ) 2 > 0. Then there is no indefinite Kähler metric on F d of constant scalar curvature whose Kähler class is proportional to 2πc 1 (L α ). In the positive-definite case, by applying Nakai s criterion for ampleness of holomorphic line bundles to L α, the following inequalities must hold (Barth et al.[2]): c 1 (L α ) [D ν0 ] = α 1 + α 3 > 0, (12) c 1 (L α ) [D ν1 ] = (α 0 + α 2 + α 3 d) (α 1 + α 3 )d > 0. Combining (11) with (12), we have (d + 1)(α 1 + α 3 ) = 2(α 0 + α 2 + α 3 d) > 2d(α 1 + α 3 ). Since α 1 + α 3 > 0, we have d + 1 > 2d, which contradicts the assumption d 1. Then c 1 (L α ) contains no positive-definite Kähler metric of constant scalar curvature. Let ω be a Kähler form (possibly indefinite) on F d. Then its Kähler class 2πη = [ω] is expressed as η = a[d ν0 ] + b[d ν1 ] for real numbers a and b. Corresponding to (6) and (8), we also obtain η 2 = b(2a bd) 0, µ = 2a b(d 2) b(2a bd). In particular, b 0 and 2a bd 0. Since F d satisfies the conditions in Proposition 3 (i.e., H 1 (F d ; O) = H 2 (F d ; O) = {0}), any Kähler form ω in the Kähler class 2πη under consideration is approximated by rational Kähler classes {ω j }. Indeed, let {ω j = 2π(a j ψ 0 + b j ψ f)} be a sequence of Kähler forms on F d satisfying that ω j ω = 2π(aψ 0 + bψ f) as j, where ψ 0 and ψ 1 are d-closed integral (1, 1)-forms that form a basis for H 1,1 (F d ; R), and where {(a j, b j )} is a sequence of pairs of rational numbers such that (a j, b j ) (a, b) as j. For each j, let m j be an integer satisfying that both m j a j and m j b j are integers and L j a holomorphic line bundle over F d with 2πc 1 (L j ) = m j [ω j ]. Then it follows from Lemma 1 that 1( F Fd (ω, τ 1 ), F Fd (ω, τ 2 )) = lim 1( F Fd (ω j, τ 1 ), F Fd (ω j, τ 2 )) = lim (b j ) 2 [b j (2a j b j d)] 3(2a j b j d) (d 2, 2d) = b2 [b (2a bd)] (d 2, 2d). 3(2a bd) 12

14 Then the normalized Bando-Calabi-Futaki obstruction for F d is given by 1( ˆFFd (η, τ 1 ), ˆF Fd (η, τ 2 )) = b(b (2a bd)) 3(2a bd) 2 (d 2, 2d). Suppose that ω is a Kähler form (possibly indefinite) on F d of constant scalar curvature in 2πη. Then ˆF Fd (η, ) 0 and hence b[b (2a bd)] = 0. By b 0, we must have 2a = (d + 1)b, which implies that η must be proportional to an integral class. However, as already examined, this is impossible, when d 1. Remark 2 Let [F] be a cohomology class corresponding to a fiber of F d = P(O(d) O) P 1 (i.e., [F] = [D ν0 ] = [D ν2 ]). Then we have b = η [F], b(2a bd) = η 2, so ˆF Fd (η, 1τ i ) is therefore expressed as follows: 1( ˆFFd (η, τ 1 ), ˆF Fd (η, τ 2 )) = (η [F])2 [ (η [F]) 2 η 2] 3(η 2 ) 2 (d 2, 2d). Note that the coefficient of (d 2, 2d) is independent of the holomorphic structure on F d. Remark 3 In the positive-definite case, Theorem 1 can also be proved by taking account of the atsushima-lichnerowicz obstruction, since the Lie algebra of all holomorphic vector fields on F d is reductive if and only if d = 0, which follows from Demazure s structure theorem (Oda [22]). However, it seems not to be known whether the atsushima-lichnerowicz obstruction is valid for the existence of indefinite Kähler metrics of constant scalar curvature. The author would like to thank Professor Helga Baum for her invitation to the program Geometry of Pseudo-Riemannian anifolds with Applications in Physics, held at the Erwin Schrödinger International Institute for athematical Physics (ESI), Vienna, He would also like to thank the ESI for hospitality during the preparation of this paper. Thanks also goes to Doctor aschler for his valuable comments on the signature of indefinite Kähler forms. References [1] S. Bando, An obstruction for Chern class forms to be harmonic, preprint (1983). [2] W. Barth, K. Hulek, C. Peters and A. Van de Ven, Compact complex surfaces, Second Enlarged Edition, Ergeb. ath. Grezgeb. (3) 4, Springer- Verlag, Berlin, Heidelberg, New York, [3] R. Bott, A residue formula for holomorphic vector fields, J. Differential Geom. 1 (1967),

15 [4] R. Bott and S. S. Chern, Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta ath. 114 (1965), [5] E. Calabi, Extremal Kähler metrics II, Differential geometry and complex analysis, , Springer, Berlin,1985. [6] J. P. D ly, Holomorphic orse inequalities, Proceedings of Symposia in Pure athematics 52, Part 2 (1991), [7] S. K, Donaldson, Scalar curvature and projective embeddings, I, J. Differential Geom. 59(2001), [8] S. K. Donaldson, Scalar curvature and stability of toric variety, J. Differential Geom. 62 (2002), [9] A. Futaki, An obstruction to the existence of Einstein Kähler metrics, Invent. ath. 73 (1983), [10] A. Futaki, On compact Kähler manifolds of constant scalar curvature, Proc. Japan Acad. Ser.A,59 (1983), [11] A. Futaki and T. abuchi, oment maps and symmetric multilinear forms associated with integral symplectic classes, Asian J. ath. 6, (2002), [12] P. Gauduchon, La 1-forme de torsion d une variété hermitienne compacte, ath. Ann. 267 (1984), [13] F. Hirzebruch, Über eine Klasse von einfach-zusammenhängenden komplexen annigfaltigkeiten, ath. Ann. 124 (1951), [14] H. Kamada, Compact scalar-flat indefinite Kähler surfaces with Hamiltonian S 1 -symmetry, Commun. ath. Phys. 254 (2005), [15] S. Kobayashi, Transformation groups in differential geometry, Ergeb. ath. Grenzgeb. 70, Springer-Verlag, Berlin, Heidelberg, New York, [16] A. Lichnerowicz, Sur les transformations analytiques des variété Kählérienne, C. R. Acad. Sci. Paris 244 (1957), [17] G. aschler, Central Kähler metrics, Trans. Amer. ath. Soc. 355 (2003), [18] G. aschler, private communication (2005). [19] Y. atsushima, Sur la structure du groupe d homéomorphismes d une certaine varieté Kählérienne, Nagoya ath. J. 11 (1957), [20] Y. atsushita, Fields of 2-planes and two kinds of almost complex structures on compact 4-dimensional manifolds, ath. Z. 207 (1991),

16 [21] Y. Nakagawa, Bando-Calabi-Futaki character of compact toric manifolds, Tohoku ath. J. 53 (2001), [22] T. Oda, Convex bodies and algebraic geometry, An introduction to the theory of toric varieties, Ergeb. ath. Grenzgeb. (3) 15, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, [23] H. Ooguri and C. Vafa, Geometry of N = 2 strings, Nuclear Phys. B 361 (1991), [24] J. Petean, Indefinite Kähler-Einstein metrics on compact complex surfaces, Comm. ath. Phys. 189 (1997), [25] G. Tian, Canonical metrics in Kähler geometry, Lectures in athematics ETH Zürich, Birkhäuser Verlag, Basel, Boston, Berlin, [26] K.P. Tod, Indefinite conformally ASD metrics on S 2 S 2, Further advances in twistor theory, Vol.III: Curved twistor spaces, 63-66, Chapman & Hall/CRC, Boca Raton, FL, Hiroyuki Kamada iyagi University of Education 149 Aramaki Aoba, Aoba-ku, Sendai , JAPAN hkamada@staff.miyakyo-u.ac.jp 15